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Author: 


Pearl,  Raymond 


Title 


Predicted  growth  of 
population  of  New  York 


Place 


New  York 

Date: 

1923 


MASTER   NEGATIVE   * 


COLUMBIA  UNIVERSITY  LIBRARIES 
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Pearl,  Raymond,  1879- 

...  Predicted  growth  of  population  of  New  York  and 
its  environs,  by  Raymond  Pearl  and  Lowell  J.  Eeed  ... 
New  York  city,  Plan  of  New  York  and  its  environs,  1923. 

42  p.  incl.  illus.  (map)  tables,  diagrs.    23*=". 
At  head  of  title:  P.  N.  Y.  4. 


X  New  York  (City)— Population.         i.  Reed,  Lowell  Jacob,  joint  author. 
II.  Plan  of  New  York  and  its  environs,    in.  Title. 


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PREDICTED  GROWTH  OF 
POPULATION  OF  NEW  YORK 
AND  ITS  ENVIRONS     ' 


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BY 
RAYMOND  PEARL 

AND 

LOWELL  J.  REED 

SCHOOL  OF  HYGIENE  AND  PUBLIC  HEALTH 
THE  JOHNS  HOPKINS  UNIVERSITY 


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PLAN  OF  NEW  YORK 
AND  ITS  ENVIRONS 

130  East  Twenty-second  Street 
New  York  City 

1923 


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COMMITTEE  ON  PLAN  of  NEW  YORK 
AND  ITS  ENVIRONS 

Frederic  A.  Delano,  Chairman 
Robert  W.  de  Forest  Dwight  W.  Morrow 

John  M.  Glenn  Frank  L.  Polk 

Frederick  P.  Keppel,  Secretary 

Flavel  Shurtleff,  Assistant  Secretary 
130  East  22d  Street,  New  York  City 

SOCIAL  SURVEY 
Shelby  M.  Harrison,  Director 


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CONTENTS 

PAGE 

Introduction 5 

Predicted  Growth  of  Population  of  New  York  and  Its  Environs 9 

Mathematical  Theory 10 

Prediction  of  Total  Population  within  the  Area  and  in  Certain  of  Its 

Subdivisions 19 

Population  Densities 24 

Population  Predictions  for  New  York  City K?*^ 

Trends  of  Certain  Elements  of  the  Population  of  New  York  City . .  (^8) 

Distribution  of  Population  by  Age  Groups 30 

Negro  Population  of  New  York  City 31 

Foreign-Born  Population  of  New  York  City 32 

Summary 34 

Tables  of  Predicted  Population  for  the  New  York  Region \^ 

Appendix.     Comparison  of  Population  Predictions  made  by  Nelson  P. 

Lewis,  of  the  Committee's  StaflF,  with  those  of  this  Study 41 


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INTRODUCTION 

SINCE  city  planning  has  to  do,  among  other  things,  with 
the  economic  and  socially  desirable  uses  of  land  areas,  it  is 
important  to  know,  at  least  approximately,  what  are  likely 
to  be  the  future  demands  of  the  people  of  a  region  for  such  areas. 
And  this  demand  is  partly  represented  in  the  population  figures— 
in  the  number  of  people  who  are  likely  to  live  in  a  given  region 
at  particular  times  in  the  future.  The  Committee  on  Plan  of 
New  York  and  Its  Environs  was  thus  early  faced  with  the  problem 
of  predicting  the  growth  of  population  in  the  New  York  Region. 

The  Committee  realized  that  this  problem  is  one  that  will  not 
be  disposed  of  at  a  single  stroke,  but  will  need  to  be  grappled  with 
in  one  form  or  another  through  all  phases  of  its  planning.  The 
beginning  point  appeared  to  be  to  compute  for  a  certain  period 
into  the  future  the  approximate  numbers  of  people  who  will  live 
in  the  New  York  Region  as  a  whole  and  in  certain  large  divisions 
of  it.  Without  waiting,  therefore,  until  its  later  studies  into  in- 
dustrial, economic,  housing,  and  other  trends,  either  general  or 
specific  for  localities,  might  indicate  factors  which  will  undoubt- 
edly modify  the  population  aggregates  for  particular  sections  of 
the  Region,  it  was  decided  to  see  what  could  be  ascertained  at 
once. 

There  are  a  number  of  methods  of  forecasting  the  populations 
of  the  future  which  are  familiar  to  students  of  the  subject,  and 
which  could  be  employed  immediately  with  data  already  avail- 
able. In  these  the  future  is  predicted  on  the  basis  of  population 
figures  of  the  past.  That  is,  the  trend  of  past  population  changes 
is  first  determined,  and  this  trend  is  then  extended  into  the  future. 
These  methods  come  down  in  general  to  two  types,  and  the 
difference  between  them  is  found  in  the  procedures  followed  in 
extending  the  curve  representing  the  future  aggregates. 

The  more  recent  of  them  is  one  developed  by  Professors  Ray- 
mond Pearl  and  Lowell  J.  Reed,  of  Johns  Hopkins  University. 
It  begins  by  indicating  several  factors  which  should  be  taken  into 


account  in  developing  a  mathematical  formula  for  predicting  the 
trend  of  future  population  growth  in  a  fixed  area:  First,  the  area 
upon  which  the  population  grows  is  finite— the  area  has  a  definite 
size  or  upper  limit.  Second,  since  population  lives  upon  limited 
areas  there  must  be  a  definite  upper  limit  to  the  number  of  persons 
who  can  live  on  that  area;  that  is,  it  is  inconceivable  that  pop- 
ulations on  particular  areas  can  increase  without  limit.  Third, 
there  is  also  a  lower  limit  to  population  which  is  zero— population 
obviously  cannot  go  below  that.  Fourth,  each  epoch  marking 
an  advance  in  human  culture  and  economy  has  made  it  possible 
for  a  given  area  to  support  more  people.  And  fifth,  the  rate  of 
growth  during  each  epoch,  in  so  far  as  it  has  been  observed,  varies, 
being  slow  at  first,  then  increasing  in  rate  to  a  maximum,  and 
then  decreasing  until  almost  a  stationary  aggregate  of  popula- 
tion is  maintained. 

With  these  factors  in  mind  Professors  Pearl  and  Reed  have 
developed  a  mathematical  equation,  aimed  to  make  the  known 
quantities  of  the  past  indicate  what  may  be  expected  in  the  future. 
In  other  words,  assuming  that  populations  cannot  grow  on  for- 
ever and  that  they  grow  at  differing  rates  at  different  periods  of 
each  epoch,  a  formula  has  been  developed  which  is  believed  to 
express  the  fundamental  law  of  normal  population  growth. 

The  Committee,  desiring  to  bring  as  much  light  as  possible  to 
bear  upon  the  local  problem  of  population  growth,  asked  Pro- 
fessors Pearl  and  Reed  to  apply  this  new  method  to  the  New  York 
Region.    The  results  obtained  are  presented  in  the  accompanying 

report. 

In  the  available  space  Professors  Pearl  and  Reed  have  confined 
themselves  to  a  description  of  their  process  and  results  without 
attempting  any  justification  of  the  method  itself.  It  may  be 
said,  however,  that  elsewhere  they  have  pointed  out  that  their 
formula  has  been  tested  by  applying  it  to  the  figures  showing 
actual  population  growth  in  a  number  of  countries,  with  the  result 
that  the  theoretical  curve  and  the  points  representing  the  re- 
corded facts  coincide  as  nearly  as  is  found  necessary  for  the  prac- 
tical confirmation  of  most  hypotheses  in  the  physical  sciences. 

It  is  realized,  however,  that  the  disturbing  factors  in  popula- 
tion trends  may  have  mugh  greater  effect  in  computing  for  smaller 
areas  than  for  those  of  whole  nations  or  countries.  That  is  to 
say,  the  introduction  of  an  unusually  large  manufacturing  in- 
dustry to  a  city,  like  the  automobile  industry  to  Detroit,  for 

6 


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example,  would  be  likely  unexpectedly  to  modify  future  popula- 
tion figures  for  a  city  or  city  region  to  a  greater  degree  than  such 
disturbance  would  for  a  whole  nation.  At  the  same  time,  while 
such  a  sudden  change  may  modify  the  course  of  the  population 
growth,  the  result  of  such  a  disturbance,  so  far  as  existing  evidence 
indicates,  would  be  merely  to  throw  the  population  trend  to  a 
new  curve  of  the  same  type  as  the  old  one  but  having  a  different 
rate. 

The  figures  presented  deal  not  only  with  the  growth  of  the 
whole  New  York  Region,  but  with  several  subdivisions  of  it. 
The  predictions  for  the  Region  as  a  whole  are  as  follows: 

Predicted  Population  (in  Round  Numbers),  New  York  Region 

Year  Number  of  Persons 

1930 11,500,000 

1940 14,100,000 

1950 16,800,000 

1960 19,600,000 

1970 22,300,000 

1980 24,800,000 

1990 27,000,000 

2000 28,800,000 

In  this  connection  it  is  interesting  and  significant  to  note  that 
the  forecasts  by  Professors  Pearl  and  Reed  for  a  considerable 
period  into  the  future— up  to  1970  at  least,  a  limit  which  provides 
ample  latitude  for  the  present  regional  planning  project— are  in 
quite  close  agreement  with  those  worked  out  by  Nelson  P.  Lewis 
as  a  part  of  his  Physical  Survey  of  the  Region  for  the  Committee, 
as  well  as  those  made  by  at  least  one  important  public  service 
corporation  which  is  also  confronted  with  the  problem  of  future 
population  trends.^  There  is  a  significant  divergence,  however, 
in  the  predictions  after  1970.  As  would  be  expected  from  the 
description  given  of  the  method  followed  by  Professors  Pearl  and 
Reed,  their  predictions  for  the  later  years  are  lower  than  those 
made  in  the  other  way.  But  even  on  this  lower,  and  in  this  sense 
more  conservative  basis,  the  growth  in  aggregate  numbers  of 
people  living  in  the  Region  promises  to  be  very  great. 

Planning  for  as  large  a  territory  as  the  New  York  Region  and 
for  a  considerable  distance  into  the  future  must  of  necessity  be 
along  broad  lines.  And  obviously,  the  longer  the  period  chosen 
the  greater  will  be  the  necessity  for  flexibility  and  adaptability 

*  A  brief  summary  of  the  estimates  made  by  Mr.  Lewis  is  given  in  the  Appen- 
dix, page  41. 

7 


in  what  is  to  be  recommended— that  is  to  say,  the  longer  ahead 
we  try  to  look  the  greater  will  be  the  necessity  of  allowing  for 
local  variations  and  adaptations  while  at  the  same  time  claiming 
the  great  benefits  which  will  come  from  unified  and  co-ordinated 
plans  for  such  a  large  Region.  This  does  not  minimize  the  great 
importance,  however,  of  getting  as  accurate  a  forecast  as  possible 
as  to  what  the  future  holds  in  store  as  to  mere  numbers  of  people 
who  will  be  living  here.  Indeed,  such  forecasts  both  for  the  near 
and  distant  future,  and  the  later  refinements  of  them,  constitute 
factors  in  planning  which  need  to  be  continually  reckoned  with. 
And  incidentally,  the  fact  that  the  predictions  by  Professors 
Pearl  and  Reed,  as  well  as  the  several  others  which  have  been 
made,  point  to  such  great  future  population  aggregates  for  this 
Region  constitutes  in  itself  one  of  the  strongest  arguments  for 
careful  and  comprehensive  planning. 

Frederick  P.  Keppel 
Secretary,  Committee  on  Plan  of  New  York  and  Its  Environs. 


8 


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PREDICTED  GROWTH  OF  POPULATION 
OF  NEW  YORK  AND  ITS  ENVIRONS 

THE  present  study  of  the  population  of  New  York  City  and 
Its  Environs  may  be  considered  as  having  two  distinct 
parts.  The  first  consists  of  a  mathematical  investigation 
into  the  growth  of  the  population  on  certain  areas  without  regard 
to  the  constitution  of  these  populations.  The  second  concerns 
certain  elements  of  the  population  and  the  growth  of  these  ele- 
ments relative  to  the  entire  population. 

Before  considering  the  results  of  the  mathematical  analysis  it 
may  be  well  to  examine  the  logic  behind  the  various  methods  of 
predicting  future  populations.  The  most  common  way  of  deter- 
mining the  probable  size  of  the  population  at  a  future  date  is  to 
determine  certain  facts  regarding  the  population  in  its  present 
state,  and  then  deduce  from  these  facts  the  size  of  the  population 
at  some  future  time.  An  illustration  of  the  application  of  this 
method  is  the  predicting  of  future  populations  from  the  present 
size  and  the  rate  of  increase  expressed  either  as  an  arithmetic  or 
geometric  rate.  It  is  well  known  that  by  this  process  we  may 
obtain  good  predictions  for  a  comparatively  short  period  of  time. 
Another  method  is  to  correlate  the  growth  of  the  population 
with  the  growth  of  some  other  variable  (usually  an  economic 
variable),  and  then  from  predictions  for  this  other  variable  to 
obtain  estimates  of  the  future  population.  This  procedure  is 
applied  when  most  of  the  population  of  a  community  is  associated 
with  some  particular  industry  which  is  known  to  be  rapidly  ex- 
panding. While  the  results  obtained  by  this  method  are  often 
accurate  over  short  intervals  of  time,  when  applied  to  longer  inter- 
vals they  are  usually  inaccurate,  for  two  reasons :  first,  the  long- 
time prediction  of  the  condition  of  any  industry  or  group  of  indus- 
tries is  itself  subject  to  error;  and  second,  no  recognition  is  made 
of  the  other  forces  acting  to  increase  and  retard  population  growth. 
A  third  method  of  predicting  future  population  rests  on  the 

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assumption  that  there  exists  a  multiplicity  of  forces  producing 
population  growth,  and  that  the  future  effects  of  these  may  best 
be  determined  by  observing  their  action  in  the  past  without 
attempting  to  differentiate  between  the  forces.  This  reduces  the 
problem  of  predicting  future  populations  to  a  mathematical-sta- 
tistical one.  It  should  be  noted  that  this  method  does  not  assume 
that  the  forces  are  constant  in  their  effect  at  different  times.  It 
merely  assumes  their  existence  and  a  continuity  in  their  action. 
The  application  of  this  method  requires  the  development  of  a 
mathematical  equation  which  will  well  represent  the  effects  of 
the  forces  in  the  past  as  exhibited  in  the  population  counts.  In 
general  this  method  of  predicting  population  is  the  most  reliable 
one  and  it  is  the  one  which  has  been  used  in  this  paper. 

MATHEMATICAL  THEORY 

The  following  account  of  the  mathematical  theory  used  in  this 
report  is  based  upon  a  paper  now  in  press  in  Metron  by  Raymond 
Pearl  and  Lowell  J.  Reed.^ 

Careful  study  of  the  matter  will  convince  one  that  at  least  the 
factors  listed  below  must  be  taken  account  of  in  any  mathema- 
tical theory  of  population  growth  which  aims  at  completeness. 
The  necessity  for  a  part  of  these  factors  is  evident  on  purely 
a  priori  grounds.  The  remainder  are  equally  obvious  and  certain 
deductions  from  observed  facts  as  to  how  populations  do  actually 

grow. 

1.  If  any  discussion  of  the  growth  of  human  population  is  to 
be  profitable  in  any  real  or  practical  sense,  ^e  area  upon  which 
the  population  grows  ^ust  be  taken  as  a  finite  one,  however  large 
its  limits.  For  the  growth  of  human  populations  the  upper  limit 
of  finite  areas  possible  of  consideration  must  plainly  be  the  habit- 
able area  of  the  earth.  Smaller  areas,  as  politically  defined  coun- 
tries, may  be  treated  each  by  itself.  But  whether  this  is  done  or 
not,  there  clearly  is  a  finite  upper  limit  of  area  on  which  human 
population  can  grow. 

2.  If  there  is  a  finite  upper  limit  to  the  area  upon  which  popula- 
tion may  grow,  then  with  equal  clearness  there  must  be  a  finite 

» See  Pearl,  R., and  Reed,  L.  J.:  On  the  Rate  of  Growth  of  the  Population  of 
the  United  States  since  1790  and  its  Mathematical  Representation.  Proceed- 
ings of  the  National  Academy  of  Sciences,  Vol.  VI,  No.  6,  June,  1920,  p.  275. 
Also  Pearl,  R.,  and  Reed,  L.  J.:  A  Further  Note  on  the  Mathematical  Theory 
of  Population  Growth.  Proceedings  of  the  National  Academy  of  Sciences,  Vol. 
Vni,  pp.  365-368. 

10 


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upper  limit  to  population  itself,  or,  in  other  words,  to  the  number 
of  persons  who  can  live  upon  that  area.  It  is  evident,  for  example, 
that  it  is  a  biological  impossibility  for  so  many  as  50,000  human 
beings  to  live,  and  derive  support  for  living,  upon  one  acre  of 
ground,  provided  every  other  acre  of  the  possibly  habitable  area 
of  the  earth  is  at  the  same  time  inhabited  to  the  same  degree  of 
density.  This  is  obviously  true  whatever  the  future  may  hold 
in  store  for  us  in  the  way  of  agricultural  discoveries,  improvements, 
or  advancements.  That  there  is  a  finite  upper  limit  to  the  popu- 
lation which  can  live  upon  a  finite  area  (as  of  the  earth)  is  really 
as  much  a  physical  as  a  biological  fact.  The  amount  of  water 
w^hich  can  be  obtained  in  a  pint  measure  is  strictly  limited  to  a 
pint.  It  cannot  by  any  chance  be  ten  gallons.  And  this  conclu- 
sion is  in  no  way  determined  or  limited  by  the  present  limitations 
of  our  knowledge  of  physics.  Nor  can  it  be  upset  by  any  future 
discoveries  to  be  made  in  the  realm  of  physics.  It  is  this  point 
which  is  so  usually  overlooked.  From  Mai  thus  to  the  present  time, 
everyone  who  has  pointed  out  that  there  must  be  some  upper 
limit-iflLjiunian_population  upon  this  globe  has  been  met  by  the 
contention  that  he  has  neglected  the  possibilities  inherent  in  the 
future  development  o(  science.  Of  course,  mture  scientific  dis- 
coveries can  have  no  bearing  upon  the  bald  fact  that  there  must 
somewhere  be  an  upper  limit  to  population.  They  can  influence 
only  the  precise  location  (or  magnitude)  of  that  upper  limit.  But 
the  discrimination  between  these  two  ideas  appears  not  to  be 
sufficiently  appreciated.  Mathematical  theories  of  population, 
even,  have  been  more  or  less  seriously  advanced  which  really 
postulated  that  with  the  passage  of  time  the  curve  of  population 
would  run  off  to  infinity.  Of  course,  attention  was  not  drawn  to 
this  feature  of  such  theories,  but  nevertheless  it  was  inherent  and 
implicit  in  them. 

3.  The  lower  limit  to  population  is  zero.  Negative  populations 
are  in  any  common,  practical  sense,  unthinkable. 

4.  History  tells  us  what  common-sense  indicates  a  priori; 
namely,  that  each  advancement  in  cultural  level  has  brought  with 
it  the  possibility  of  additional  population  growth  within  any  de- 
fined area.  In  the  hunting  stage  of  human  culture  the  number  of 
persons  who  can  be  supported  upon  a  given  area  is  small.  In  the 
pastoral  stage  of  culture  more  persons  can  subsist  upon  a  given 
area,  though  the  absolute  number  is  still  small.  In  the  general 
agricultural  stage  of  civilization  the  possibilities  of  population 

II 


♦r 


I 


per  unit  of  area  become  again  enhanced.  The  commercial  and 
industrial  stages  of  culture  permit  great  increases  of  population, 
provided,  of  course  (and  only  under  this  condition),  that  there 
still  remain  somewhere  else  less  densely  populated  areas  where 
the  means  of  subsistence  can  be  produced  in  excess  of  local  needs. 
In  other  words,  each  geographical  unit  which  has  been  inhabited 
for  any  long  time  has,  so  far  as  the  evidence  available  indicates, 
had  a  succession  of  waves  of  eras  of  population  growth,  each  su- 
perposed upon  the  last,  and  each  marking  the  duration  of  a  more 
or  less  definite  cultural  epoch. 

5.  Within  each  cultural  epoch  or  cycle  of  population  growth 
the  rate  of  growth  of  population  has  not  been  constant  in  time. 
Instead,  the  following  course  of  events  has  apparently  occurred 
generally,  and  indeed  almost  universally.  At  first  the  population 
grows  slowly,  but  the  rate  constantly  increases  to  a  certain  point 
where  it,  the  rate  of  growth,  reaches  a  maximum.  This  point  may 
presumably  be  taken  to  represent  the  optimum  relation  between 
numbers  of  people  and  the  subsistence  resources  of  the  defined 
area.  This  point  of  maximum  rate  of  growth  is  the  point  of  inflec- 
tion of  the  population  growth  curve.  After  that  point  is  passed 
the  rate  of  growth  becomes  progressively  slower,  until  finally  the 
curve  stretches  along  nearly  horizontal,  in  close  approach  to  the 
upper  asymptote,  or  limit,  which  belongs  to  the  particular  cultural 
epoch  and  area  involved. 

All  these  factors  must  certainly  be  taken  account  of  in  a  mathe- 
matical theory  of  population  growth.  For  convenience  they  may 
be  recapitulated  in  brief  as  follows: 

1.  Finite  limit  of  area. 

2.  Upper  limiting  asymptote  of  population. 

3.  Lower  limiting  asymptote  of  population  at  0. 

4.  Epochal  or  cyclical  character  of  growth,  successive 

cycles  being  additive. 

5.  General  shape  of  curve  of  growth. 

With  these  fundamental  postulates  in  mind  we  may  now  proceed 
to  their  mathematical  expression  and  generalization.  In  our  first 
paper^  we  took  as  a  first  approximation  to  the  law  expressing 
normal  population  growth,  equation  (I). 

b 


.   y       e  -ax  -\-c 

1  Pearl,  R.,  and  Reed,  L.  J.    Loc.  cit. 

12 


(I) 


^ 


< 


l      ( 


This  satisfied  perfectly  postulates  1,  2,  3,  and  in  a  fair  degree  5. 
It  made  no  attempt  to  satisfy  4.  Since  our  first  paper  was  pub- 
lished we  have  learned  that  nearly  three-quarters  of  a  century  ago 
a  Belgian  mathematician,  Verhulst,^  in  two  long  since  forgotten 
papers,  which  appear  never  to  have  been  generally  recognized  in 
the  later  literature  on  population,  anticipated  us  in  the  use  of  equa- 
tion (I)  to  represent  population  growth.  The  only  recent  writer 
on  the  subject  who  seems  to  have  known  of  Verhulst's  work  is 
DuPasquier,*  who  himself  makes  use  of  a  slight  and,  as  it  seems 
to  us,  entirely  unjustified  and  in  practice  usually  incorrect  modi- 
fication of  (I).  There  have,  of  course,  been  many  attempts  at 
getting  mathematical  expressions  of  population  growth,  or  of 
growth  in  general.  We  shall  make  no  attempt  to  review  all  this 
literature,  chiefly  for  the  reason  that  most  of  the  mathematical 
expressions  brought  forward  have  been  wholly  lacking  in  gener- 
ality. They  have  been  special  curves,  doctored  up  with  greater 
or  less  skill  to  fit  a  particular  set  of  observations,  often  involving 
assumptions  which  could  not  possibly  hold  in  any  general  law  of 
growth.  A  recent  paper  by  Lehfeldt*  develops  an  idea  as  to 
the  changes  of  a  variable  in  time  which,  fundamentally,  seems 
to  be  similar  to  that  set  forth  in  the  present  paper.    He  says : 

"Let  q  be  the  quantity  whose  changes  in  time  t  are  to  be 
studied.  It  is  not  to  be  expected  that  the  changes  of  q  itself  should 
be  symmetrical  in  time,  for  all  the  changes  observed  in  the  later 
half  of  the  period  of  change  refer  to  values  of  q  larger — possibly 
many  times  larger — than  in  the  earlier  half.  But  log  q  may  very 
possibly  undergo  symmetrical  changes,  so  we  will  assume  that  it 
is  a  'normal  error  function'  of  the  time,  i.  e., 

log  q  =  log  qo  +  kF  (|) 

where  q©  is  the  value  of  q  at  a  certain  moment  (the  'epoch'): 
t  is  the  time  in  years  before  or  after  the  epoch :  T  is  a  constant 

period  and      F  (x)  =  —7=  1     e  dx      and  k  is  a  constant." 

VirJo 

*  Verhulst,  P.  F. :  Recherches  Mathematiques  sur  la  Loi  d'  Accroissement  de 
la  Population.    Mem.  de  I'Acad.  Roy.  de  Bruxelles.  T.  XVIII,  pp.  1-58,  1844. 

Idem.  Deuxieme  Memoire  sur  la  Loi  d 'Accroissement  de  la  Population. 
Ibid.  T.  XX,  pp.  1-52,  1846. 

*  DuPasquier,  L.  G.:  Esquisse  d'une  Nouvelle  Theorie  de  la  Population. 
Vierteljahrschr.  der  Naturforsch.  Ges.,  Zurich,  Jahrg.  63,  pp.  236-249,  1918. 

'  Lehfeldt,  R.  A.:  The  Normal  Law  of  Progress.  Journal  Royal  Statistical 
Society,  Vol.  LXXIX,  pp.  329-332,  1916. 

13 


I 


't 


!'• 


II 


It  seems  to  us  that,  mathematically,  this  method  of  approaching 
the  problem  is  much  less  general,  and  much  more  difficult  of 
application  and  of  interpretation  than  our  treatment  of  the  prob- 
lem which  follows. 
Considered  generally,  the  curve 

y       e  -ax  +c 

may  be  written 

k  (II) 


y  = 


where 


1  +  me  ka'  X 
1 


k  =  -;      m  =  -;      and  ka'  =  —a. 
c  c 


Now  the  rate  of  change  of  y  with  respect  to  x  is  given  by 

^  =  -a'y  (k-y) 


or 


dy 
dx 


y  (k-y) 


=  -  a' 


(III) 


If  y  be  the  variable  changing  with  time  x  (in  our  case  popula- 
tion), equation  (III)  amounts  to  the  assumption  that  the  time 
rate  of  change  of  y  varies  directly  as  y  and  as  (k— y),  the  constant 
k  being  the  upper  limit  of  growth,  or  in  other  words  the  value  of 
the  growing  variable  y  at  infinite  time.  Now  since  the  rate  of 
growth  of  y  is  dependent  upon  factors  that  vary  with  time  we 
may  generalize  (III)  by  using  f  (x)  in  place  of  —a',  f  (x)  being 
some  as  yet  undefined  function  of  time. 


Then 


whence 


and 


where 


dy 


y  (k-y) 


=  f  (x)  dx. 


y  = 


h^Zl  =  e  -k/f  (x)dx 
my 

k  k 


1  +  me  -  k/f  (x)  dx      1  4-  me  F  (x) 
F(x)  =  -kjf  (x)dx 


(IV) 


Then  the  assumption  that  the  rate  of  growth  of  the  dependent 
variable  varies  as  (a)  that  variable,  (b)  a  constant  minus  that 
variable,  and  (c)  an  arbitrary  function  of  time,  leads  to  equation 

14 


t 


i,     r 


(IV),  which  is  of  the  same  form  as  (I),  except  that  ax  has  been 
replaced  by  F  (x).  If  now  we  assume  that  f  (x)  may  be  repre- 
sented by  a  Taylor  series,  we  have 

k 


y  = 


If 


1  +  me  aix  +  ajx^  -f  aaJ^  -h anx" 

a»  =  as  =  a4  =  an  =  0 


(V) 


then  (V)  becomes  the  same  as  (I). 

If  m  becomes  negative  the  curve  becomes  discontinuous  at  finite 
time.  Since  this  cannot  occur  in  the  case  of  the  growth  of  the 
organism  or  of  populations  nor,  indeed,  so  far  as  we  are  able  to 
see,  for  any  phenomenal  changes  with  time,  we  shall  restrict  our 
further  consideration  of  the  equation  to  positive  values  only  of  m. 
Also  since  negative  values  of  k  would  give  negative  values  of  y, 
which  in  the  case  of  population  or  individual  growth  are  unthink- 
able, we  shall  limit  k  to  positive  values. 

With  these  limitations  as  to  values  of  m  and  k  we  have  the 
following  general  facts  as  to  the  form  of  (V).  y  can  never  be 
negative  (i.  e.,  less  than  zero),  nor  greater  than  k.  Thus  the 
complete  curve  always  falls  between  the  x  axis  and  a  line  parallel 
to  it  at  a  distance  k  above  it.  Further  we  have  the  following 
relations : 

If      F  (x)  =    00  y  ===  0 

F  (x)  =  -  00      y  =  k 


F  (x)  =  -  0     y  =  T-r— 
^  ^  "^       1  +  m 

k 


F  (x)  =  +  0      y  = 


1  +  m 


from  below 
from  above 


dy 

The  maximum  and  minimum  points  of  (V)  occur  where  j~  =  0« 


But 


5^  =  y(k-y)  F(x); 


therefore  we  have  maximum  and  minimum  points  wherever 
F'  (x)  =  0. 

dx 
The  fact  that  -r-  =  0  when  either  y  =  Oory  —  k  =  0  shows 

dy 
that  the  curve  passes  oflF  to  infinity  asymptotic  to  the  lines  y  =  0 

and  y  =  k. 

15 


I 


I6 


S5 
O 

o 

(^ 

O 
> 

S 

o 
H 

u 

H 

Ptf 
O 
fa 

> 

U 

o 


r  CO 

I' 

i| 

•o  "  o 

O  3W 
^  Wo 

0«fi  3 
O  0;a 

rt  Si  « 

ega 

•""  Ml 


^^3^? 


D 

o 


V  0)  s 

jcj:  9 
•s  a>  o 

S'o  a 


y  > 


a 

3 


•3   3   « 


0) 


J3  — 


0 


>: 


The  points  of  inflection  of  (V)  are  determined  by  the  inter- 
sections of  (V)  with  the  curve 


y  =  o- 


k      k  F'^Cx) 


2      2  F'(x)2 


(VI) 


Since  we  are  seldom  justified  in  using  over  five  arbitrary  con- 
stants in  any  practical  problem,  we  may  limit  equation  (V)  still 
further  by  stopping  at  the  third  power  of  x.  This  gives  the  equa- 
tion 

y  =  7-1 ^ — nr^  (VH) 

^        1  +  me  aix  +  ajx*  +  ajx*  ^        ' 

If  an  is  positive,  the  curve  of  equation  (V)  is  reversed  and  be- 
comes asymptotic  to  the  line  A  B  at  x  =  —  00  and  to  the  x  axis 
at  X  =  +  00 .  Thus  in  equation  (VII)  as  negative  is  a  case  of 
growth,  and  as  positive  is  a  case  of  decay. 

Equation  (VII)  has  several  special  forms  that  are  of  interest, 
among  them  being  a  form  similar  in  shape  to  the  autocatalytic 
curve  (i.  e.,  with  no  maximum  or  minimum  points  and  only  one 
point  of  inflection),  except  that  it  is  free  from  the  two  restrictive 
features  mentioned  in  our  first  paper,  that  is,  location  of  the  point 
of  inflection  in  the  middle  and  symmetry  of  the  two  limbs  of  the 
curve.  Asymmetrical  or  skew  curves  of  this  sort  can  only  arise 
when  equation  (V)  has  no  real  roots.  While  any  odd  value  of  n 
may  yield  this  form  of  curve  the  simplest  equation  that  will  do 
it  is  that  in  which  n  =  3,  so  that  the  equation  of  this  curve  be- 
comes that  of  (VII). 

Having  determined  that  the  growth  within  any  one  epoch  or 
cycle  may  be  approximately  represented  by  equation  (I),  or  more 
accurately  by  (VII)  the  next  question  is  that  of  treating  several 
epochs  or  cycles.  Theoretically  some  form  of  (V)  may  be  found 
by  sufficient  labor  in  the  adjustment  of  constants  so  that  one 
equation,  with  say  5  or  7  constants,  would  describe  a  long  history 
of  growth  involving  several  cycles.  Practically,  however,  we 
have  found  it  easier  and  just  as  satisfactory  in  other  respects  to 
treat  each  cycle  by  itself.  Since  the  cycles  of  any  case  of  growth 
are  additive  we  may  use  for  any  single  cycle  the  equation 

k 


y  =  d  + 


or  more  generally 


y  =  d  + 


1  +  me  ka'x 
k 


(VIII) 


1  +  me  aix  +  ajx*  +  aix* 
17 


MILLIONS 


1850 


MILLIONS 


1850 


MILLIONS 
5 


1850 


20 

15 
10 

^^^ 

5 

^^^.^..r-^ 

AREA  I 

0 

r— *"                     ' 

.*v. 


1900 


1950 


2000 


Z050 


ZIOO 


1900 


1950 


2000 


2050 


2100 


2050 


2100 


Diagram  2.    Population  Curves  for  Areas  I,  II,  and  III 

Small  circles  indicate  observed  iK>pulations,  by  which  the  curves  are  deter- 
mined. Horizontal  lines  mark  the  upper  asymptotes  or  limits  of  the  respec- 
tive curves. 

I8 


^ 


> 


^ 
I 


where,  in  both  forms,  d  represents  the  total  growth  attained  in 
all  the  previous  cycles.  The  term  d  is,  therefore,  the  lower  asymp- 
tote of  the  cycle  of  growth  under  consideration  and  d  +  k  is  its 
upper  asymptote. 

In  treating  any  two  adjacent  cycles  it  should  be  noted  that 
the  lower  asymptote  of  the  second  cycle  is  frequently  below  the 
upper  asymptote  of  the  first  cycle  due  to  the  fact  that  the  second 
cycle  is  often  started  before  the  first  one  has  had  time  to  reach 
its  natural  level.  This,  for  instance,  would  be  the  case  where  a 
population  entered  upon  an  industrial  era  before  the  country  had 
reached  the  limit  of  population  possible  under  purely  agricultural 
conditions. 

The  mathematical  theory  herein  developed  has  been  applied 
by  the  authors  to  the  populations  of  all  the  large  countries  of  the 
world  and  to  several  of  the  large  cities.  In  all  of  these  cases  the 
agreement  between  the  observed  values  and  those  of  the  equation 
is  of  the  same  order  as  that  usually  found  in  the  application  of  a 
mathematical  law  to  measurements  in  the  field  of  science. 

PREDICTION   OF   TOTAL   POPULATION   WITHIN   THE   AREA 
AND  IN  CERTAIN  OF  ITS  SUBDIVISIONS 

The  area  under  consideration  is  shown  in  the  map  preceding 
this  report  and  is  comprised  of  the  following  counties: 

New  York:  New  York,  Kings,  Queens,  Bronx,  Richmond, 
Nassau,  Suffolk,  Westchester,  Putnam,  Dutchess  (part),  Rock- 
land, Orange  (part). 

New  Jersey:  Hudson,  Bergen,  Essex,  Union,  Passaic,  Morris, 
Somerset,  Middlesex,  Monmouth  (part). 

Connecticut:  Fairfield  (part). 

This  total  area  is  subdivided  as  indicated  on  the  map  into  three 
areas  defined  as  follows : 

Area  I.  New  York  City,  Hudson  County,  Newark. 

Area  II.  Nassau  County,  Westchester  County,  part  of  Fair- 
field County,  Essex  County  (excluding  Newark),  part  of  Bergen 
County,  part  of  Passaic  County,  Union  County. 

Area  III.  The  remainder  of  the  total  area. 

19 


20 


c 


.1 

CO    <2 

<  8. 


g 


o 


CO  0) 

5  3 
o  u 

Bp 

» 


04 

O 
to 

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z 
o 


h 

s  a 

0)  >« 

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2     ;5 


^-^  Irt  0) 

J  -"  4) 

O  jj.a 

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(o;S 
Si 

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S 

a> 
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»» 


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A 


> 


1 


The  following  table  shows  the  population  in  thousands  for  each 
of  the  areas,  as  compiled  from  the  United  States  census  reports: 

Table  1.— Recorded  Population  of  the  New  York  Region  (in  Thousands) 


Year 

Area  I 

Area  II 

Area  III 

Total  area 

1790 

225 

1800 

290 

1810 

359 

1820 

428 

1830 

560 

1840 

764 

1850 

758 

135 

270 

1.163 

1860 

1,311 

210 

314 

1.835 

1870 

1,714 

280 

381 

2,375 

1880 

2,237 

349 

440 

3,026 

1890 

2,966 

477 

523 

3,966 

1900 

4,074 

683 

628 

5,385 

1910 

5,651 

1,029 

787 

7.467 

1920 

6,664 

1.383 

932 

8,979 

The  method  used  in  predicting  the  future  populations  of  these 
areas  is  that  described  in  the  preceding  section.  The  actual  pro- 
cedure was  to  fit  the  equation  y  =  d  H ^^   ,     to  the  popu- 

e  ~  ^x  -|-c 

lation  counts  of  the  four  areas.  In  the  case  of  the  total  area 
and  of  Area  I  the  term  d  was  found  to  be  so  small  that  it  could 
be  neglected.    Thus  the  actual  curve  used  in  these  areas  was 

y  =  ; — .    The  four  curves  adjusted  by  the  method  of 

•^         e  -  ax  -l-c 

least  squares  to  the  populations  in  question  lead  to  the  following 
equations,  in  which  y  represents  population  in  thousands,  and  x 
represents  time  in  years  since  1800. 

256.0527 


Total  area 


Area  I 


Area  II 


y      e  -  .032300X  -f  .0073368 

156.2838 

y  -"  e  -  .0349348X  +  .0078578 

8.77755 


y  =  87  +  — : 


.0428817X  -I-  .0008207 
30.3083 


Area  III        y  -  150  +  ^  _  .029018lx  +  .0074311 

Since  the  accuracy  of  any  mathematical  prediction  depends  in 
part  on  the  degree  to  which  the  curve  satisfies  the  original  observa- 

21 


. ~  '*>  » 


» 

H 

prt 

< 


2 
o 

e 

H 
O 

5^0 


n 

O 


en 


g 


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o 

1-4 

Pi 
< 

O 


o 

put 

IS 
o 

u 

o 
a* 
Pi 


H 

O 


-       H 


P< 
O 
to 


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o 

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o 

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22 


tions,  it  is  of  interest  to  compare  the  observed  values  of  the  popu- 
lation with  the  values  obtained  from  the  equations.  Table  7,  in 
the  series  of  tables  at  the  end  of  the  report,  shows  the  computed 
values  for  the  populations  of  the  different  areas  at  ten-year  inter- 
vals from  1790  to  2100  with  the  observed  values  for  the  years 
through  1920.  The  relationship  of  the  observed  population  to 
the  curves  may  also  be  seen  in  Diagrams  1  and  2.  Examination 
of  the  table  and  of  these  diagrams  shows  that  the  curves  do  rep- 
resent the  growth  of  the  population  throughout  the  observed 
period  with  a  high  degree  of  accuracy.  As  a  further  check  on  the 
precision  of  the  curves  the  root  mean  squared  deviation  of  the 
observed  values  from  the  curves  was  computed,  and  in  no  case 
was  the  root  mean  squared  deviation  over  4  per  cent  of  the  aver- 
age population. 

Diagrams  3  and  4  are  presented  in  order  that  the  growth  of  the 
populations  of  the  different  areas  may  be  compared.  Diagram  3 
is  on  arithmetic  scale  and  may  be  used  to  compare  the  propor- 
tion of  the  population  in  the  different  areas  at  any  particular 
instant  of  time.  Diagram  4  is  on  logarithmic  scale  and  shows  by 
the  slopes  of  the  curves  the  varying  rates  of  growth  of  the  popu- 
lations of  the  different  areas.  Comparison  of  these  slopes  shows 
that,  whereas  up  to  1880  Area  I  had  the  most  rapidly  growing 
population,  after  that  time  Area  II  increased  at  the  most  rapid 
rate.  The  rate  for  Area  II  continues  to  be  most  rapid  until  about 
the  year  2000,  when  the  rate  for  Area  III  exceeds  it. 

The  percentage  distributions  of  the  total  population  by  areas 
are  given  in  Table  8  at  the  end  of  the  report,  and  are  plotted  in 
Diagram  5.  Area  I,  through  its  rapid  growth  in  population  from 
1850  to  1900,  reached  a  position  where  it  contained  three-fourths 
of  the  population  of  the  entire  area.  Since  1900,  however,  the 
percentage  of  the  total  population  in  Area  I  has  been  declining, 
while  that  in  Area  II  has  been  rising.  This  movement  continues 
in  the  predicted  population  until  about  the  year  2050,  when  these 
two  areas  stabilize.  Area  I  containing  about  57  per  cent  and  Area 
11,31  per  cent  of  the  total  population.  Area  III  has  lost  ground 
in  comparison  with  the  other  areas  throughout  the  observed  por- 
tion of  the  curve.  Starting  at  a  value  of  21  per  cent  in  1850,  the 
percentage  has  decreased  steadily  since  that  time  to  a  little  more 
than  10  in  1920.  It  continues  to  decrease  in  the  predicted  por- 
tion of  the  curve  to  a  low  point  of  9.53  in  1960,  after  which  it 
rises  slightly  and  stabilizes  at  about  12  per  cent. 

23 


I 


PER  CCNT 
100 


FCRSONS  PCR  SQUARE  MILE 
100,000 


50^000 


10,000 


5.000 


„ 


t 


/■ 


1850 


1900 


1950 


2000 


2050 


2100 


Diagram  5.  Percentage  Curves  Showing  Change  in  the  Propor- 
tion OF  THE  Total  Population  Falling  in  Each  of  the  Three  Subdi- 
visions OF  THE  Region 


POPULATION  DENSITIES 

From  the  predicted  population  figures  we  may  obtain  values 
for  the  predicted  densities  of  population.  Table  9,  following  the 
report,  shows  these  densities  for  the  three  sub-areas  and  for  the 
total  area.  It  should  be  noted  that,  although  there  is  a  slight 
movement  toward  an  equality  of  density  throughout  the  area, 
this  movement  falls  far  short  of  realization.  The  plot  of  these 
density  figures  on  logarithmic  paper  in  Diagram  6  shows  that 
although  the  rate  of  increase  of  density  in  Area  II  is  much  greater 
than  that  in  Area  I,  in  absolute  value  the  density  of  Area  II  is 
always  much  less  than  that  of  Area  I. 

24 


i 


1,000 


500 


too 


1850 


1900 


1950 


2000 


2050 


2100 


Diagram  6.    Population  Density  Curves  Obtained  from  the  Predicted 
Populations  for  the  Total  and  Component  Areas 

As  in  the  previous  diagrams,  the  prominent  horizontal  lines  indicate  the 
upper  asymptotes  of  the  respective  curves,  and  here  mark  the  predicted  limits 
of  population  density  for  each  area. 


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The  changes  that  are  taking  place  in  the  densities  are  perhaps 
made  more  clear  by  the  schematic  plots  shown  in  Diagrams  7 
and  8.  The  vertical  scales  in  these  diagrams  represent  density 
in  persons  per  square  mile.  The  segments  of  the  base  represent 
the  diameters  of  concentric  circles.  The  line  AA'  is  proportional 
to  the  diameter  of  a  circle  whose  area  is  equal  to  Area  I ;  the  line 
BB'  to  the  diameter  of  a  circle  with  an  area  equal  to  the  sum  of 
Areas  I  and  II ;  and  the  line  CC  to  the  diameter  of  the  total  area. 

Plotting  above  these  segments  columns  whose  tops  indicate 
the  respective  densities,  we  have  a  picture  of  the  piling  up  of  the 
population  on  these  areas.  In  Diagram  7,  two  such  density  pic- 
tures have  been  made,  one  for  the  year  1880,  and  the  other  for  the 
asymptotic  conditions  of  the  population,  which  are  approximately 
the  conditions  as  predicted  for  the  year  2100.  The  striking  fea- 
ture of  this  diagram  is  the  enormous  density  of  the  central  area 
for  the  limiting  value  of  the  population. 

Diagram  8  shows  a  similar  plot  on  logarithmic  background  and 
is  of  especial  interest  in  bringing  out  the  point  that  great  as  the 
density  of  the  central  area  is  in  the  asymptotic  limits,  we  find 
this  area  less  dense,  relative  to  the  other  areas,  than  in  the  year 
1880. 

POPULATION  PREDICTIONS  FOR  NEW  YORK  CITY 

Although  the  population  figures  of  Area  I  are  nearly  the  same 
as  those  of  New  York  City  proper,  it  seemed  advisable  to  deter- 
mine a  population  curve  for  the  area  now  included  in  New  York 
City.  The  Census  Bulletin  for  1920  gives  the  following  figures 
for  the  population  of  the  present  area  of  New  York  City: 

Table  2.— Recorded  Population  of  Present  Area  of  New  York  City 

(in  Thousands) 


Year 

Population 

Year 

Population 

1850 
1860 
1870 
1880 

696 
1,175 
1,478 
1,912 

1890 
1900 
1910 
1920 

2.507 
3,437 
4,767 
5,620  ^-^ 

^^1. 

i>  ^>7 


26 


27 


Applying  the  methods  used  for  the  other  areas  we  determine  the 
following  population  curve  for  New  York  City: 

138.017 


MILLIONS 
16 


y  = 


e-.034935x +.008971 


In  this  curve  as  in  the  others  x  represents  time  in  years  measured 
from  the  year  1800  as  an  origin,  and  y  represents  population  in 
thousands.  This  curve  is  plotted  in  Diagram  9,  together  with 
the  values  of  the  observed  population.  The  accuracy  with  which 
the  curve  represents  the  observed  values  is  obviously  on  a  par 
with  that  of  the  previous  cases.  The  values  of  the  predicted  popu- 
lation at  ten-year  intervals  are  given  in  the  last  column  of  Table 
10.  The  proportion  of  the  predicted  population  of  Area  I  that 
falls  within  New  York  City  is  shown  in  the  following  table: 

Table  3.— Predicted  Population  of  Area  I  and  of  Present  New  York 

City 


Year 

Predicted  population 
(in  thousands) 

Per  cent  of 
population  of  Area  I 

Area  I 

New  York  City 

in  New  York  City 

1850 
1900 
1950 
2000 
2050 
2100 

857 
4,086 
11,878 
17,797 
19,489 
19,818 

753 

3,506 

9,672 

13,948 

15,105 

15.337 

87.86 
85.81 
81.43 
78.37 
77.51 
77.39 

Asymptote 

19,889 

15,385 

77.35 

The  steady  decline  of  these  percentages  shows  the  tendency 
toward  a  uniform  density  of  population  within  Area  I,  which  fact 
is  in  keeping  with  the  findings  of  the  previous  section  with  regard 
to  density  throughout  the  entire  area. 

TRENDS  OF  CERTAIN  ELEMENTS  OF  THE  POPULATION  OF 

NEW  YORK  CITY 

In  considering  the  changes  that  take  place  in  the  size  of  a  single 
element  of  a  population,  two  procedures  are  possible.  We  may 
consider  the  element  of  the  total  population  as  a  distinct  universe 
and  fit  to  it  the  same  type  of  equation  used  for  the  total  popula- 

28 


V 


'( 


M 


1890 


1900 


1950 


2000 


2050 


2100 


Diagram  9.    Population  Curve  for  New  York  City 
The  area  included  is  that  of  the  city  as  at  present  constituted.    Small  circles 
indicate  observed  population,  and  a  horizontal  line  marks  the  asymptote  of 
the  curve  and  the  predicted  limit  of  population. 

tion,  or  we  may  form  the  ratios  of  the  particular  element  to  the 
total  population  ak  different  instants  of  time  and  then  determine 
the  trend  of  these  ratios.  The  latter  procedure  will  in  general  be 
the  better  of  the  two.  These  ratios  for  certain  elements  of  the 
population  remain  so  nearly  constant  that  there  is  no  evidence 
of  a  trend  during  the  period  for  which  we  have  observations.  In 
these  cases  the  average  ratio  over  the  known  period  should  be 
obtained  and  applied  to  the  predicted  values  of  the  total  popula- 
tion. In  case  there  is  evidence  of  a  trend  in  the  ratios,  the  ques- 
tion arises  as  to  what  type  of  curve  is  to  bie  used  to  represent  this 
trend.  A  consideration  of  the  general  forms  used  to  represent 
population  growth  shows  that  the  ratios  of  two  such  curves  may 

be  well   represented   by   the  equation,  y    =   d   +      _  ^^  , 

Thus,  in  an  actual  example,  we  may  fit  the  above  equation  to  the 
known  ratios,  compute  ratios  at  future  times,  and  by  applying 
these  to  the  predicted  values  of  the  total  population  obtain  the 
size  of  the  population  element.  We  shall  now  consider  certain 
elements  of  population  and  shall  determine  their  probable  future 

29 


I 


values.  It  should  be  noted  that  the  probable  errors  of  the  pre- 
dicted values  are  greater  in  the  case  of  the  elements  of  the  popu- 
lation than  in  the  case  of  the  population  as  a  whole. 

DISTRIBUTION  OF  POPULATION  BY  AGE  GROUPS 
The  following  table  shows  the  population  of  New  York  City 
for  certain  age  groups,  stated  in  actual  numbers  and  also  as  per- 
centages of  the  total  population. 

Table  4.— Recorded  Population  of  New  York  City  by  Age  Groups 


Age 
group 

Years 

Population 

Per  cent  of  total  population 

1890 

1900 

1910 

1920 

1890 

1900 

1910 

1920 

0-  4 
5-  9 
10-14 
15-19 
20-44 
45  and 
over 

164,686 
140,026 
130,651 
148,843 
686,825 

240,652 

397,287 
354,747 
301,264 
302,751 
1,530,239 

545,478 

507,080 
438,263 
422,431 
457,616 
2,145,583 

789,108 

560,869 
536,490 
494,867 
453,758 
2,488,415 

1,077,844 

10.894 
9.263 
8.643 
9.846 

45.434 

15.920 

11.577 

10.337 

8.779 

8.822 

44.590 

15.895 

10.653 
9.207 
8.874 
9.614 

45.074 

16.578 

9.994 
9.559 
8.818 
8.085 
44.339 

19.205 

Total 

1,511,683 

3,431,766 

4,760,081 

5,612,243 

100.000 

100.000 

100.000 

100.000 

Examination  of  the  table  shows  no  appreciable  trend  in  the  per- 
centages of  the  different  age  groups.  This  being  the  case  we  may 
take  the  average  of  these  percentages  as  representing  the  propor- 
tions of  the  total  population  to  be  found  in  the  different  age  groups. 
These  average  percentages  are : 

Age  group  Per  cent  of 

Years  total  population 

0-  4 10.779 

5-9 9.592 

10-14 8.778 

15-19 9.092 

20-44       44.859 

45  and  over 16.900 

Applying  these  percentages  to  the  values  of  the  predicted  popu- 
lation of  New  York  City  we  find  probable  values  for  the  popula- 
tion within  each  age  group.  These  values  are  given  in  Table  10 
at  ten-year  intervals  from  1920  to  2100. 

Other  elements  of  population  for  which  the  statistics  available 

30 


show  no  evidence  of  trend  may  be  treated  by  the  method  used 
for  age  distributions. 

NEGRO  POPULATION  OF  NEW  YORK  CITY 
Many  racial  groups  within  the  total  population  have  rates  of 
growth  that  are  greater  than  that  of  the  population  taken  as  a 
whole.  The  Negro  element  within  New  York  City  is  an  illustra- 
tion of  this  fact.  The  size  of  the  Negro  population  and  its  per- 
centage of  the  total  is  shown  in  the  following  table: 

Table  5.-Recorded  Total  and  Negro  Population  of  New  York  City 


Year 


1860 
1870 
1880 
1890 
1900 
1910 
1920 


Total 
population 


813,669 
942,292 
1,206,299 
1,515,301 
3,437,202 
4,766,883 
5,620,048 


Negro 
population 


12,574 
13,072 
19,663 
23,601 
60,666 
91,709 
152,467 


Per  cent 
of  total 


1.55 
1.39 
1.63 
1.56 
1.76 
1.92 
2.71 


The  figures  for  the  first  four  years  are  based  on  the  old  area  of 
New  York  City,  since  for  these  years  the  numbers  of  Negroes 
within  the  present  city  limits  are  not  available.  However,  as  the 
curve  is  to  be  based  on  percentages,  this  difference  can  have  little 

or  no  effect. 

Applying  to  these  percentages  the  equation 

y  =  d  +  e-ax  +  c 
we  obtain  the  following  expression  for  the  percentage  of  Negroes 

in  the  total  population 

.0003567 
y  =  1.409  +  g  _  .07094X  +  .00009641 

In  this  equation  y  represents  the  percentage  of  the  total  popu- 
lation that  is  Negro,  and  x  represents  the  time  measured  in  years 
from  1800  as  an  origin.  From  this  equation  we  derive  the  values 
of  the  future  Negro  population  given  in  Table  11  and  plotted  in 
Diagram  10.  It  should  be  noted  that  the  observed  values  for 
the  years  1860,  1870,  1880,  and  1890  have  been  adjusted  to  the 
area  within  the  present  city  limits.    The  equation  shows  that  the 

31 


THOUSANDS 
800 


«00 


200 


i 


400  — 


1850        1900        1950       2000       2050        2100 

Diagram  10.    Prediction  Curve  for  Negro  Population  of  New  York 

City 
Small  circles  indicate  observed  population,  and  a  horizontal  line  marks  the 
asymptote  of  the  curve  and  the  predicted  limit  of  Negro  population. 

Negro  population  is  increasing  toward  a  position  where  in  the 
asymptotic  conditions  it  will  comprise  5  per  cent  of  the  total 
population. 

FOREIGN-BORN  POPULATION  OF  NEW  YORK  CITY 
The  foreign-born  population  of  New  York  City  is  an  illustra- 
tion of  a  population  element  whose  rate  of  growth  is  less  than 
that  of  the  total  population.    The  actual  numbers  of  foreign  born 
are  increasing  steadily,  but,  as  we  see  from  Table  6,  the  pro- 

Table  6.— Recorded  Total  and  Foreign-Born  Population  of  New 

York  City 


Year 

Total 

Foreign-born 

Per  cent 

population 

population 

of  total 

1870 

942,292 

419,094 

44.48 

1880 

1.206,299 

478,670 

39.68 

1890 

1,515,301 

639,943 

42.23 

1900 

3,437,202 

1,260,918 

36.68 

1910 

4,766,883 

1,927,703 

40.44 

1920 

5,620,048 

1,991,547 

35.44 

32 


Y 


( 


> 


portion  of  foreign  born  in  the  total  population  of  the  city  is  de- 
clining. 

As  in  the  case  of  the  Negroes,  the  percentages  from  1870  to 
1890,  inclusive,  are  computed  from  the  populations  of  the  old  city. 
While  the  movement  of  these  percentages  has  not  been  as  steady 
as  in  the  case  of  the  Negroes,  there  has  evidently  been  within  the 
past  fifty  years  a  decrease  in  the  proportion  of  the  population 
that  is  foreign  born. 

The  curve  obtained  from  these  percentages  is 


88.5143 


y  =  9.91  +  g  _  .0071962X  +  .982510 


']S 


This  curve  leads  to  the  predicted  values  for  the  foreign-born  popu- 
lation that  are  given  in  Table  1 1  and  plotted  in  Diagram  1 1 .  This 
curve  brings  out  very  clearly  the  fact  that  continuation  of  the 
present  decrease  in  the  percentage  of  foreign  born  in  the  popula- 


MILLIONS 

4 


1850 


1900 


1950        2000        2050 


2100 


2150         2200        2250 


Diagram  11. 


Prediction  Curve  for  Foreign-born  Population  of  New 
York  City 

The  circles  represent  observed  population,  and  the  horizontal  line  which 
the  curve  approaches  indicates  the  upper  asymptote  of  the  curve.  Ip  thiscase 
the  rate  of  ^owth  becomes  negative  after  the  year  2000,  and  the  limit  of  popu- 
lation  is  predicted  at  about  three-quarters  of  the  present  number  of  foreign 
born  in  the  city. 

33 


tion  will  lead  to  a  time  when  the  actual  numbers  of  foreign  born 
will  begin  to  decrease.  This  fact  may  seem  strange  in  the  light 
of  the  large  increase  in  our  foreign-born  population  at  the  present 
time,  but  it  is  quite  in  harmony  with  the  present  immigration 
policies  of  the  country. 

The  prediction  furnished  by  the  curve  is  that  about  eighty 
years  from  now  New  York  City  will  contain  its  maximum  num- 
bers of  foreign  born,  after  which  time  the  foreign-born  popula- 
tion will  decrease  until  it  reaches  a  limiting  position  at  about  a 
million  and  a  half.  In  considering  this  forecast  it  should  be  real- 
ized that  any  prediction  of  the  actual  number  of  foreign-born 
population  of  the  city  must  possess  much  less  reliability  than  the 
earlier  predictions  of  this  report.  On  the  one  hand  the  observed 
values  are  too  few  to  be  fully  satisfactory  for  the  purpose  of  pre- 
diction and,  as  this  element  of  the  population  has  undergone  less 
regular  increase,  the  observed  points  show  greater  variation  from 
the  predicted  curve  than  in  the  other  cases.  In  the  case  of  the 
foreign-born  population  it  is  evident  that  single  incidental  fac- 
tors have  much  greater  effect  upon  the  rate  of  growth  than  in 
any  of  the  other  cases  considered. 


SUMMARY 

Perhaps  the  best  way  to  appreciate  the  great  increase  in  popu- 
lation that  may  be  expected  to  take  place  in  the  area  under  dis- 
cussion is  to  take  a  cross-section  view  of  the  population  as  pre- 
dicted for  some  specific  date,  for  example,  the  year  2000.  At 
that  time  the  population  of  the  total  area  will  have  increased 
from  its  present  value  of  9,000,000  to  about  29,000,000;  that  is, 
the  area  will  contain  more  than  three  times  its  present  popula- 
tion. This  increase  is  not  evenly  distributed  throughout  the  area, 
however,  for  we  find  Area  I  with  2.62  times  its  present  popula- 
tion, while  the  present  populations  of  Area  II  and  Area  III  are 
multiplied  by  6.24  and  3.23,  respectively.  It  is  therefore  evident 
that  Area  II  makes  a  far  greater  relative  gain  than  either  of  the 
other  two  areas. 

In  density.  Area  I  will  have  reached  by  the  year  2000  the  value 
of  48,759  persons  per  square  mile,  which  is  about  the  present 
density  of  Manhattan  and  Bronx  boroughs.  Area  II  will  contain 
7,507  persons  per  square  mile.  Its  density  will  be  not  quite  half 
the  present  density  of  Area  I.    Area  III  will  not  be  very  densely 

34 


,>- 


f 


populated  by  the  year  2000,  its  density  at  that  time  being  764 
persons  per  square  mile,  or  about  that  of  Area  II  in  1910. 

The  distribution  of  population  by  age  will  be  about  the  same 
in  the  year  2000  as  at  present,  so  that  such  groups  as  children  of 
school  age,  persons  of  voting  age,  etc.,  will  be  increased  over  their 
present  value  by  the  same  factors  as  those  previously  given  for 
the  general  population. 

The  Negro  population  will  have  tripled  and  will  constitute  5 
per  cent  of  the  total  population,  whereas  at  present  they  form 
but  2.6  per  cent  of  the  total. 

The  foreign-born  population  will  have  increased  from  its 
present  value  of  2,080,000  persons  to  approximately  3,750,000, 
which  will  be  about  the  peak  for  this  element  of  the  population. 
Expressed  as  a  percentage  of  the  total  population,  the  foreign 
born  in  the  year  2000  will  be  less  than  at  present,  the  percentage 
at  that  time  being  26.9  as  against  36.3  at  present. 

According  to  the  prediction  equations  the  population  situation 
in  the  year  2000  will  be  very  near  that  at  which  the  population 
will  tend  to  stabilize,  the  one  outstanding  exception  being  that 
the  number  of  foreign  born  in  the  population  will  ultimately  tend 
to  stabilize  at  about  1,525,000,  which  is  far  below  the  number 
predicted  for  the  year  2000. 


i 


35 


TABLES  OF  PREDICTED  POPULATION  FOR  THE  NEW  YORK 

REGION 


.M^l 


Table  7.— Predicted  and  Observed  Populations   (in  Thousands)  of 
Total  Area  and  Three  Subordinate  Areas 


* 

Area  I    ^ 

Area  II 

Area  III 

Total  area 

Year 

Pre- 

Ob- 

Pre- 

Ob- 

Pre- 

Ob- 

Pre- 

Ob- 

dicted 

served 

dicted 

served 

dicted 

served 

dicted 

served 

1790 

•  • 

•  • 

•  • 

184 

225 

1800 

254 

290 

1810 

350 

359 

1820 

482 

428 

1830 

662 

560 

1840 

908 

764 

1850 

857 

758 

161 

135 

275 

270 

1.242 

1,163 

1860 

1,195 

1,311 

201 

210 

316 

314 

1,692 

1,835 

1870 

1,652 

1,714 

261 

280 

369 

381 

2,295 

2,375 

1880 

2,265 

2,237 

351 

349 

437 

440 

3,092 

3,026 

1890 

3,067 

2,966 

488 

477 

525 

523 

4,131 

3,966 

1900 

4,086 

4,074 

690 

683 

636 

628 

5,460 

5,385 

1910 

5,336 

5,651 

986 

1,029 

775 

787 

7,li7 

7,467 

1920 

6,803 

6,664 

1,408 

1,383 

944 

932 

9,122 

8,979 

1930 

8,441 

1,990 

1,146 

,   , 

11,458 

1940 

10,166 

2,754 

1,380 

14,066 

1950 

11,878 

3,700 

1,643 

16,841 

1960 

13,479 

4,785 

1,924 

19,647 

1970 

14,895 

5,927 

2,221 

22,342 

1980 

16,086 

7,025 

2.514 

24,806 

1990 

17,047 

7.993 

2,794 

26,958 

.. 

2000 

17,797 

8,783 

3,051 

28,765 

2010 

18,366 

9,390 

3,278 

30,232 

2020 

18,790 

9,832 

3.473 

31,391 

2030 

19,102 

10,144 

3.636 

32,287 

2040 

19,327 

10,358 

3.768 

32,968 

s 

2050 

19,489 

10,502 

3,874 

33,480 

2060 

19,606 

10,598 

3,958 

33,860 

2070 

19,688 

10,661 

4,022 

34,141 

2080 

19,747 

10,703 

4.072 

34,347 

2090 

19,789 

10,730 

4,111 

34,498 

2100 

19,818 

10,748 

4.140 

34,608 

Asymptote 

19,889 

•   • 

10,782 

■   • 

4,229 

•   • 

34,900 

«   • 

Table  8.— Percentage  Distribution  of  Predicted  Population,  by  Areas 


Year 

Area  I 

Area  II 

Area  III 

1850 

66.28, 

12.45 

21.27 

1860 

69.80 

11.74 

18.46 

1870 

72.39 

11.44 

16.17 

1880 

74.19 

11.50 

14.31 

1890 

75.17 

11.96 

12.87 

1900 

75.50 

12.75 

11.75 

1910 

75.19 

13.89 

10.92 

1920 

74.31 

15.38 

10.31 

1930 

72.91 

17.19 

9.90 

1940 

71.09 

19.26 

9.65 

1950 

68.97 

21.49 

9.54 

1960 

66.77 

23.70 

9.53 

1970 

64.64 

25.72 

9.64 

1980 

62.78 

27.41 

9.81 

1990 

61.24 

28.72 

10.04 

2000 

60.06 

29.64 

10.30 

2010 

59.18 

30.26 

10.56 

2020 

58.55 

30.63 

10.82 

2030 

58.09 

30.85 

11.06 

2040 

57.78 

30.96 

11.26 

2050 

57.55 

31.01 

11.44 

2060 

57.39 

31.02 

11.59 

2070 

57.28 

31.02 

11.70 

2080 

57.20 

31.00 

11.80 

2090 

57.14 

30.99 

11.87 

2100 

57.10 

30.97 

11.93 

Asymptote 

56.99 

30.89 

12.12 

\i 


36 


37 


>C^Q.GLy^     r 


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y 


Table  9.— Predicted  Density  of  Population  (Persons  per  Square  Mile) 
OF  Total  Area  and  Three  Subordinate  Areas 


Year 

Area  I 

Area  II 

Area  III 

Total  area 

1790 

•  • 

33 

1800 

46 

1810 

•  • 

63 

1820 

«  • 

87 

1830 

•  • 

120 

1840 

•  • 

164 

1850 

2,348 

138 

69 

225 

1860 

3,274 

172 

79 

306 

1870 

4,526 

223 

92 

415 

1880 

6,205 

300 

109 

559 

1890 

8,403 

417 

131 

747 

1900 

11,194 

590 

159 

988 

1910 

14,619 

843 

194 

1,287 

1920 

18,638 

1,203 

236 

1,650 

1930 

23,126 

1,701 

287 

2,073 

1940 

27,852 

2,354 

346 

2,545 

1950 

32,542 

3,162 

411 

3,046 

1960 

36,929 

4,090 

482 

3,554 

1970 

40,808 

5,066 

556 

4,042 

1980 

44,071 

6,004 

630 

4,487 

1990 

46,704 

6,832 

700 

4,877 

2000 

48,759 

7,507 

764 

5,204 

2010 

50,318 

8,026 

821 

5,469 

2020 

51,479 

8,403 

870 

5,679 

2030 

52,334 

8,670 

911 

5,841 

2040 

52,951 

8,853 

944 

5,964 

2050 

53,395 

8,976 

970 

6,056 

2060 

53,715 

9,058 

991 

6,125 

2070 

53,940 

9,112 

1,007 

6,176 

2080 

54,101 

9,148 

1,020 

6,213 

2090 

54,216 

9,171 

1,030 

6,241 

2100 

54,296 

9,186 

1,037 

6,260 

Asymptote 

54,490 

9,215 

1,059 

6,313 

Table  10.— Predicted  Age  Distribution  of  Population  of  New  York 

City  (in  Thousands) 


'V 


Years  of  age 

Total 

Year 

0-4 

5-9 

10-14 

15-19 

20-44 

45  and 
over 

1920 

618 

550 

503 

521 

2,571 

968 

5,731 

1930 

758 

675 

617 

639 

3,155 

1,188 

7,032 

1940 

902 

803 

735 

761 

3,756 

1,415 

8,372 

1950 

1,043 

928 

849 

879 

4,339 

1,634 

9,672 

1960 

1,171 

1,042 

953 

988 

4,872 

1,835 

10,861 

1970 

1,282 

1,141 

1,044 

1,081 

5,335 

2,009 

11,892 

1980 

1,374 

1,222 

1,119 

1,159 

5,717 

2,154 

12,745 

1990 

1,447 

1,288 

1,178 

1,220 

6,022 

2,269 

13,424 

2000 

1,504 

1,338 

1,224 

1,268 

6,257 

2,357 

13,948 

2010 

1,546 

1,376 

1,259 

1,304 

6,435 

2,424 

14,344 

2020 

1,577 

1,404 

1,285 

1,331 

6,566 

2,473 

14,636 

2030 

1,601 

1,424 

1,304 

1,350 

6,661 

2,509 

14,849 

2040 

1,617 

1,439 

1,317 

1,364 

6,730 

2,536 

15,003 

2050 

1,628 

1,449 

1,326 

1,373 

6,776 

2,553 

15,105 

2060 

1,638 

1,457 

1,334 

1,381 

6,815 

2,567 

15,192 

2070 

1,644 

1,463 

1,338 

1,386 

6,841 

2,577 

15,249 

2080 

1,648 

1,466 

1,342 

1,390 

6,859 

2,584 

15,289 

2090 

1,651 

1,469 

1,345 

1,393 

6,871 

2,588 

15,317 

2100 

1,653 

1,471 

1,346 

1,395 

6,880 

2,592 

15,337 

Asymptote 

1,658 

1,476 

1,351 

1,399 

6,902 

2,599 

15,385 

38 


I 


1^ 


39 


APPENDIX 


Table  11.— Predicted  Negro  and  Foreign-Born  Population 

OF  New 

York  City 

Total 

Negro 

Per  cent 
Negro 

Foreign-born 

Per  cent 

Year 

population 

population 

population 

foreign 

(in  t  lousands) 

(in  thousands) 

(in  thousands) 

born 

1860 

1,046 

15 

1.434 

471 

45.00 

1870 

1,443 

21 

1.459 

627 

43.47 

1880 

1,969 

30 

1.510 

826 

41.97 

1890 

2,650 

43 

1.609 

1,073 

40.50 

1900 

3,506 

63 

1.794 

1,369 

39.06 

1910 

4,539 

96 

2.117 

1,709 

37.66 

1920 

5,731 

150 

2.609 

2,080 

36.30 

1930 

7,032 

228 

3.236 

2,460 

34.98 

1940 

8,372 

324 

3.869 

2,821 

33.70 

1950 

9,672 

423 

4.373 

3,140 

32.46 

1960 

10,861 

511 

4.707 

3,395 

31.26 

1970 

11,892 

583 

4.899 

3,581 

30.11 

1980 

12,745 

638 

5.003 

3,697 

29.01 

1990 

13,424 

679 

5.056 

3,752 

27.95 

2000 

13,948 

709 

5.083 

3,756 

26.93 

2010 

14,344 

731 

5.096 

3,724 

25.96 

2020 

14,636 

747 

5.102 

3,663 

25.03 

2030 

14,849 

758 

5.106 

3,586 

24.15 

2040 

15,003 

766 

5.107 

3,497 

23.31 

2050 

15,105 

772 

5.108 

3,400 

22.51 

2060 

15,192 

776 

5.108 

3,304 

21.75 

2070 

15,249 

779 

5.109 

3,207 

21.03 

2080 

15,289 

781 

5.109 

3,110 

20.34 

2090 

15,317 

783 

5.109 

3,017 

19.70 

2100 

15,337 

784 

5.109 

2,928 

19.09 

Asymptote 

15,385 

786 

5.109 

1,525 

9.91 

I 


COMPARISON    OF    POPULATION    PREDICTIONS    MADE    BY 
NELSON    P.    LEWIS,   OF    THE    COMMITTEE'S    STAFF,  WITH 

THOSE  OF  THIS  STUDY 

IN  connection  with  other  investigations  carried  on  by  the  Com- 
mittee, Nelson  P.  Lewis,  Director  of  the  Physical  Survey, 
made  a  series  of  population  predictions  for  the  total  area  and 
for  the  three  subdivisions  of  it  which  were  later  used  by  Pro- 
fessors Pearl  and  Reed  in  the  present  study.  Mr.  Lewis's  esti- 
mates are  shown  in  the  diagram  and  table  reproduced  below. 
The  method  followed  consisted  in  determining  the  rate  of  in- 
crease of  population  by  decades  from  1850  to  1920  and  then, 
with  this  as  a  basis,  estimating  empirically  the  trend  up  to  the 
year  2000.  In  doing  this  he  recognized  the  fact  that  the 
population  of  the  Region  has  been  increasing  by  a  series  of  de- 

MILLI0N5 
50.0 


^ 


r 

i-. 


i 


1 


1650 


1900 


950 


2000 


40 


Population  Curves  for  the  New  York  Region  and  the  Three  Com- 
ponent Areas,  Showing  Actual  Increases  from  1850  to  1920,  and  Esti- 
mated  Future  Growth  up  to  the  Year  2000 

41 


creasing  percentages,  and  it  was  assumed  that  this  would  con- 
tinue. 

It  will  be  observed,  as  suggested  in  the  Introduction,  that  the 
predictions  made  in  these  ways  for  the  whole  area  run  very  close 
together  for  the  next  fifty  years.  This  is  true  to  a  considerable 
degree  also  for  the  smaller  divisions  of  the  Region,  as  will  be  seen 
by  further  comparison  of  the  diagram  shown  here  and  the  one 
presented  on  page  22.  The  difference  between  the  figures  is  in  no 
case  more  than  4  per  cent  until  1970.  From  1970  upward  the 
divergence  becomes  greater,  the  figures  of  Mr.  Lewis  being  25  per 
cent  higher  than  those  of  Professors  Pearl  and  Reed  when  the 
year  2000  is  reached. 

Comparison  of  Population  Predictions  for  the  Total  New  York  Region 


Predicted 

Predicted 

Per  cent  of 

Year 

by  Professors 
Pearl  and  Reed 

by  Mr.  Lewis 

difference 

1930 

11,500,000 

11,000,000 

-    4 

1940 

14,100,000 

13,800,000 

-    2 

1950 

16,800,000 

16,600,000 

-    1 

1960 

19,600,000 

20,000,000 

-f-    2 

1970 

22,300,000 

24,000,000 

+    8 

1980 

24,800,000 

28,000,000 

4-  13 

1990 

27,000,000 

32,000,000 

-h  19 

2000 

28,800,000 

36,000,000 

+  25 

T 


42 


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END  OF 
TITLE 


